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“application” in Latin

The English word application itself goes back to Latin plicō, through applicātiōnem.

Riddle & Arnold A Copious and Critical English–Latin Lexicon, 1864

application
use, usus: usurpatio (act of using in a particular instance): a bad application, abusus. To admit of a wider application, latius patere.
Application of mind, animi attentio (Cic.). oftener intentio (act of directing the thoughts to anything): (diligentia (care with which one attends to anything: opposed to indiligentia). In Cic. applicatio animi is the attaching of the mind to an object with affection. To make a practical application of any thing, aliquid ita tractare, ut id ad usum transferas. All the applications of an art, omnia, quae aliqua arte effici possunt.
Application (= particular use) of a word, must be translated by circumlocution with verbo uti (not verbum usurpare, adhibere): subjicere sententiam vocabulo; vocabulo aliquid significare, declarare. To use a word in a rare application, verbum doctiuscule ponere. “Cic., too, makes a similar application of the word,” item consimiliter Cic. verbo isto utitur. “Cic. uses the word in a contrary application,” contra valet cum Cic. - ita dicit. Vid: Positio dictionis is not Latin.
Petition, Vid:
Act of applying, circumlocution by participle admotus. “By the application of herbs,” admotis herbis (animam admotis fugientem sustinet herbis, Ov.).
To make the application (e.g., of a tale), interpretari aliquid.

Quick answer

What is the Latin word for application?
The usual Latin words for “application” are ūtor, ūsūrpātiō, admoveō; also abūtor, adhibeō, dēclārō. The right choice depends on the sense; see the distinctions below.

Entry from J. E. Riddle & T. K. Arnold, A Copious and Critical English–Latin Lexicon (rev. C. Anthon, New York: Harper, 1864), digitised by Fergus J. P. Walsh, CC BY-SA 4.0; Latin markup adapted by latindictionary.co.